Building Pyramids

The solid generated by this process is a cube, the quadrilateral is a square, and the smallest internal angle is 90 degrees.

This is easier seen going backward, starting with a cube and cutting off four pyramid-shaped corners.

But if you don't see it, you can start with the tetrahedron and grow pyramids as suggested. The resulting quadrilateral has to be a diamond, due to symmetry. So proving that both diagonals are the same length will prove that it is a square. The above is a projection of the tetrahedron into the plane of its base. All lengths and angles below are in this plane. The length of the side of the tetrahedron is arbitrarily picked to be 2. $x$ is from a right triangle given by $sin(60)=\frac{1}{2x}$ , therefore $x=\frac{1}{2\times sin(60)}$ . $y$ is from another right triangle $y=x\times tan(60)=\frac{1}{2\times cos(60)}=1$ . $y$ is half of the diagonal, so the entire diagonal is 2. (Since $y$ is horizontal, it is not distorted.) The other diagonal, $2x$ in the image, is distorted, but we know it's length. It is the length of the side of the tetrahedron, therefore also 2.

This would have created a polyhedron of 8 vertices, 12 edges, and 6 faces. What could that be? A cube, if it possesses tetrahedral symmetry.